Following [1]}, let \((\Omega , {{F}},\mathsf {P})\)  be a probability space. A \(\mathcal {H}\) -valued random variable is a measurable function \(X:\Omega \rightarrow \mathcal {H}\) , where \(\mathcal {H}\)  is endowed with the Borel \(\sigma \) -algebra. We denote by \(\sigma (X)\)  the \(\sigma \) -field generated by \(X\) . The expectation of a random variable \(X\)  is denoted by \(\mathsf {E}[X]\) . The conditional expectation of \(X\)  given a \(\sigma \) -field \({{A}} \subset {{F}}\)  is denoted by \(\mathsf {E}[X|{{A}}]\) . A \(\mathcal {H}\) -valued random process is a sequence \((x_n)\)  of \(\mathcal {H}\) -valued random variables. The abbreviation a.s. stands for 'almost surely'.
